Network Biology- part IV

Jun Zhu, Ph. D.

Professor of Genomics and Genetic Sciences

Icahn Institute of Genomics and Multi-scale Biology

The Tisch Cancer Institute

Icahn Medical School at Mount Sinai

New York, NY

@IcahnInstitute

http://research.mssm.edu/integrative-network-biology/

Email: jun.zhu@mssm.edu

Association

networks

Probabilistic causal

networks

Biological details revealed

Data required to train models

Biological networks/pathways

1. How do genes in the same

module interact?

2. How do genes in different

modules interact?

3. Can we make causal

inferences to elucidate

signaling pathway for

disease targets?

4267 top genes in BxH liver female rescan qtl overlap (num(p(GGC)<1e-15)>100 ~abs(cor)>0.5886)

What are Bayesian networks? Association vs Causality

From Stephen Friend

A simple biological question: are there

causal/reactive relationships?

A Bayesian network approach:

Best model

What are Bayesian networks?

• A Bayesian network is an expert system that captures

all existing knowledge;

• They are also called belief networks, Bayesian belief

networks, causal probabilistic networks;

• A Bayesian network consists of

• a directed acyclic graph (a set of nodes and directed edges

connecting nodes)--DAG

• A set of conditional probability tables (for discrete data) or

probability density functions (for continuous data)

A Bayesian network

C

A B

F

D

( | , )p C A B

(D | )p B

E (E | )p B

(F | C)p

DAG Conditional

probability tables

Bayesian network

• A tree is a Bayesian network

C

A

B

F

D E

Bayesian network

C

A B

F

D E

• A Bayesian network is not a tree

Bayesian network

• Conventional Notations

( ) ( | ( ))i i

i

p p A pa AA

1 2 n{A ,A , ,A } are nodes.A

( ) is the joint probability of nodes .p A A

( ) are parent nodesof .i ipa A A

Bayesian network

C B

A

D E

• A diverging structure

out-degree =4

Bayesian network

D

A B C

• A converging structure

in-degree =3

Bayesian network

• Why a DAG is required?

( ) ( | ( ))i i

i

p p A pa AA

• It is guaranteed that there is a node Aj in a DAG that

has no child.

j

j

j

( ) ( \ { }) ( | \ { })

( \ { }) ( | ( ))

( ( | ( )))* ( | ( ))

j j

j j

i i j

i j

p p A p A A

p A p A pa A

p A pa A p A pa A

A A A

A

Bayesian network: usages

• Bayesian networks can be used to predict outcomes

or diagnose causal effects (if structures are known)

• Bayesian networks can be used to discover causal

relationships (if structures are not known)

Bayesian network: an example

Alarm

burglar Earthquake

Phonecall

Radio Internet

• A burglar alarm system

Bayesian network: a classifier

• What is a naïve Bayes net

C B

A

D E

( (A)

(( | , , )

(

p A,B,C,D,E)= p(B | A)p(C | A)p(D | A)p(E | A)p

p A,B,C,D,E)p A B C D

p B,C,D,E)

Bayesian network

B

A

C

• How to train a Bayesian network

A=a1 A=a2 A=a3

B=b1 7 12 25

B=b2 20 30 28

B=b3 25 20 6

A=a1 A=a2 A=a3

C=c1 15 8 20

C=c2 11 25 18

C=c3 27 10 16

Bayesian network

B

A

C

• How to construct a Bayesian network? Enumerating

possible structures

B

A

C B

A

C B

A

C

B

A

C B

A

C B

A

C

Bayesian network

• How to construct a Bayesian network? Enumerating all

possible structures is impossible

,NN N is thenumberof nodes

Bayesian network

• How to construct a Bayesian network? Heuristic approach

xi

Pa1 Pa2 Pan

xi

Pa1 Pa2 Pan

xi

Pa1 Pa2 Pan

xi

Pa1 Pa2 Pan

X Pan+1 Paj

a b c

Bayesian network

• How to construct a Bayesian network? Heuristic approach

D

A B

Parameters to estimate=3x3x3

D

A B C

Parameters to estimate=3x3x3x3

Bayesian network

• How to construct a Bayesian network? Heuristic approach

( | ) ( )( | )

( )

p D M p Mp M D

p D

ˆBIC 2ln ( ) ln( )

: number of samples

: number of parameters toestimate

p D | M k n

n

k

Bayesian network

• How to construct a Bayesian network? averaging

Zhu et al., PLoS CompBio, 2007

Zhu et al., Nature Genetics, 2008

Bayesian network • How to construct a Bayesian network? Enforcing DAG

after averaging

1. Calculate shortest distance

2. Identify loops

3. Remove the weakest link in a loop

4. Go to step 1

Zhu et al., PLoS CompBio, 2007

Bayesian network

• How to construct a Bayesian network? Upper limit on

in-degree

Parameters to estimate=

xi

Pa1 Pa2 Pan

13n

Bayesian network

• Continuous vs discrete models

• Discrete model is faster, easier to capture high order

interactions

• Any discretization lost information

Bayesian network

• Missing information

B

A

C B

X

C

A

B

X

C

A

Bayesian network

• Biological network is context specific

• Bayesian network is just a snapshot under a specific

condition

Other ways to infer causal networks

• Boolean network, Graphic Gaussian model

• Conditional Mutual Information

• ODE model

• Structural equation

modeling by differential equations

uxfdtdx )(/

Response function

Observed states

Perturbation

Gardner et al, Science, 2003

modeling by ordinary differential

equations (ODE)

▶ Assume static state dx/dt=0

▶ Assume linear relationships f(x)=Ax

0uAx

regulatory matrix

response matrix

01 uAx

ODE: advantages and disadvantages

▶ Advantages:

• Simple

• Can model feedback loops

▶ Disadvantages:

• Need large amount of data

• Need even more data to capture non-linear

relationships

Aknowledgements Mount Sinai

Genomics Institute

Eric Schadt

Bin Zhang

Zhidong Tu

Charles Powell

Patrizia Casaccia

Zhu lab

Seungyeul Yoo

Eunjee Lee

Li Wang

Luan Lin

Quan Long

•Icahn Institute of Genomics and Multiscale Biology,

Icahn School of Medicine at Mount Sinai

•Janssen

•Canary Foundation

•Prostate Cancer Foundation

•NIH

•NCI

Supported by:

Boston University

Avrum Spira

Joshua Campbell

U Washington

Roger Baumgarner

Berkerley

Rachel Brem

Princeton

Lenoid Kruglyak